Stokes Law Calculator
Stokes law gives the drag force on a small sphere moving slowly through a viscous fluid. It is the foundation of sedimentation theory and particle size analysis.
💧 Fluid Dynamics📐 F = 6πηrv☁️ Sedimentation
F = 6πηrv
Fviscous drag forceN
ηdynamic viscosityPa·s
rsphere radiusm
vrelative velocitym/s
Sphere Radius (m)
Velocity (m/s)
Fluid Viscosity (Pa·s)
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Drag Force | F | F = 6πηrv | N |
| Terminal Velocity | vt | vt = 2r²(ρp−ρf)g/(9η) | m/s |
| Radius | r | r = F/(6πηv) | m |
| Viscosity | η | Dynamic viscosity | Pa·s |
Step-by-Step Examples
Example 1
Microsphere in Water
A sphere of radius 1 μm moves through water (η=0.001 Pa·s) at 0.0001 m/s.
- F = 6π × 0.001 × 1×10⁻⁶ × 1×10⁻⁴
- F = 1.88×10⁻¹² N; Re based on diameter ≈ 2×10⁻⁷
✓ 1.88 pN — safely inside the creeping-flow regime
Example 2
Oil Droplet Sedimentation
Oil droplet r=10 μm (rho_p=900), water (rho_f=1000, eta=0.001), g=9.8.
- v_t = 2(10⁻⁵)²(900-1000)×9.8/(9×0.001)
- v_t = -2.18×10⁻⁵ m/s
✓ 0.0218 mm/s upward (buoyant, rises)
Example 3
Fine Sand Particle
A fine spherical particle with r=10 μm and density 2650 kg/m³ settles in water.
- v_t = 2(10⁻⁵)²(2650-1000)×9.8/(9×0.001)
- v_t = 3.59×10⁻⁴ m/s; Re ≈ 0.007
✓ 0.359 mm/s — Stokes approximation is self-consistent
Real-World Applications
Particle Size Analysis
Sedigraph instruments use Stokes settling velocity to measure particle size distributions.
Aerosols and Fine Droplets
Stokes drag models sufficiently small droplets and particles; ordinary raindrops usually require finite-Re drag correlations.
Sedimentation Analysis
Stokes settling provides an idealized starting point for particle-size and suspension analysis; concentrated or non-spherical particles need corrections.
Oil Refining
Separating oil droplets from water in centrifuges uses Stokes law scaling with centrifugal g.
Common Mistakes to Avoid
⚠️
Applying to large/fast objects
Stokes law requires particle Re = ρv(2r)/η ≪ 1 (creeping flow). For larger Re, drag coefficient corrections are needed.
⚠️
Forgetting 6 (not 3 or 4)
The constant is 6pi, not 3pi or 4pi. It comes from the full Navier-Stokes solution for a sphere.
⚠️
Ignoring buoyancy in terminal velocity
Terminal velocity formula includes (rho_p - rho_f): subtract fluid density, not just use particle density.
Connected Formulas
validity gate: Re ≪ 1 →Reynolds Numberbalances weight and buoyancy at →Terminal Velocityreplaced at higher Re by →Drag Force
Frequently Asked Questions
What is creeping flow? ▾
Flow around a sphere where inertia is negligible compared to viscous forces (Re < 1). The velocity field is symmetric front-to-back, and Stokes law is exact.
How does Stokes law change at higher speeds? ▾
At Re > 1, wake forms behind sphere. Use drag coefficient: F = 0.5*Cd*rho*v²*pi*r². Stokes gives Cd = 24/Re.
What is the Millikan oil drop experiment? ▾
Used Stokes drag to measure electron charge. Oil droplets fell/rose under gravity and electric field; Stokes drag balanced them at terminal velocity.
Can Stokes law apply to non-spherical particles? ▾
Yes, with a shape factor correction. Elongated particles experience more drag than spheres of equal volume.
What is the Einstein relation for diffusion? ▾
D = kT/(6*pi*eta*r), the Stokes-Einstein equation. Links diffusion coefficient to particle radius and viscosity.
How does a centrifuge improve sedimentation? ▾
Replacing g with centrifugal acceleration a = omega²*r increases effective g by thousands of times, dramatically speeding Stokes settling.
What is hindered settling? ▾
At high particle concentrations, particles interfere with each other, reducing effective settling velocity below Stokes prediction.
Why does Stokes drag depend linearly on velocity? ▾
At low Re, fluid deforms viscously around the sphere. Viscous force is proportional to velocity gradient, which scales linearly with particle speed.